№ 4 Найти АС

Найдем угол A.

$$∠A = 180^{\circ} - ∠B - ∠C = 180^{\circ} - 45^{\circ} - 105^{\circ} = 30^{\circ}$$

Применим теорему синусов:

$$\frac{AC}{\sin B} = \frac{BC}{\sin A}$$ $$\frac{AC}{\sin 45^{\circ}} = \frac{6\sqrt{2}}{\sin 30^{\circ}}$$ $$AC = \frac{6\sqrt{2} \cdot \sin 45^{\circ}}{\sin 30^{\circ}} = \frac{6\sqrt{2} \cdot \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 6 \sqrt{2} \cdot \sqrt{2} = 6 \cdot 2 = 12$$

Ответ: AC = 12